Review Lecture - Water waves and their development in space and time

1985 ◽  
Vol 400 (1818) ◽  
pp. 1-18 ◽  
Keyword(s):  
Wave Propagation ◽  
Shallow Water ◽  
Deep Water ◽  
Water Waves ◽  
Water Wave ◽  
Hydraulic Jump ◽  
Soliton Solutions ◽  
Nonlinear Effects ◽  
Wave Fields ◽  
Turbulent Bore

Most practical predictions of water-wave propagation use linear approximations based on the concepts of ‘geometric’ rays and group velocity. Although this is successful, or adequate, in many instances, there are phenomena that can only be fully understood in terms of nonlinear effects. The recent boom in soliton-related studies has shed much light on the nonlinear aspects of wave propagation in shallow water. However, for waves on deeper water some of the nonlinear effects are only now being appreciated. A few, such as the focusing pattern of steady wave fields have direct parallels in shallow water; while others, such as deep-water soliton solutions, have their own rich structure. In deep or shallow water, wavebreaking is the most eye-catching development of a wave field. With the exception of the classical turbulent bore or hydraulic jump, our present models are still some way from giving a quantitative appreciation of important effects such as energy dissipation and momentum transfer, but causes of breaking for deep-water waves are now a little better understood.

scholarly journals New multi-soliton solutions of Whitham-Broer-Kaup shallow-water-wave equations

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 137-144 ◽  
Author(s):  
Sheng Zhang ◽  
Mingying Liu ◽  
Bo Xu
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Bilinear Forms ◽  
Shallow Water Waves ◽  
Bilinear Method ◽  
Shallow Water Wave ◽  
Shallow Water Wave Equations

In this paper, new and more general Whitham-Broer-Kaup equations which can describe the propagation of shallow-water waves are exactly solved in the framework of Hirota?s bilinear method and new multi-soliton solutions are obtained. To be specific, the Whitham-Broer-Kaup equations are first reduced into Ablowitz- Kaup-Newell-Segur equations. With the help of this equations, bilinear forms of the Whitham-Broer-Kaup equations are then derived. Based on the derived bilinear forms, new one-soliton solutions, two-soliton solutions, three-soliton solutions, and the uniform formulae of n-soliton solutions are finally obtained. It is shown that adopting the bilinear forms without loss of generality play a key role in obtaining these new multi-soliton solutions.

Multi-Layer Modeling of Wave Groups From Deep to Shallow Water

2003 ◽  
Author(s):  
Patrick Lynett ◽  
Philip L.-F. Liu ◽  
Hwung-Hweng Hwung ◽  
Wen-Son Ching
Keyword(s):  
Shallow Water ◽  
Deep Water ◽  
Water Waves ◽  
Water Wave ◽  
Linear Wave ◽  
Layer Model ◽  
Wave Groups ◽  
Good Linear ◽  
Model Equations ◽  
Linear Accuracy

A set of model equations for water wave propagation is derived by piecewise integration of the primitive equations of motion through N arbitrary layers. Within each layer, an independent velocity profile is determined. With N separate velocity profiles, matched at the interfaces of the layers, the resulting set of equations have N+1 free parameters, allowing for an optimization with known analytical properties of water waves. The optimized two-layer model equations show good linear wave characteristics up to kh ≈8, while the second-order nonlinear behavior is well captured to kh ≈6. The three-layer model shows good linear accuracy to kh ≈14, and the four layer to kh ≈20. A numerical algorithm for solving the model equations is developed and tested against nonlinear deep-water wave-group experiments, where the kh of the carrier wave in deep water is around 6. The experiments are set up such that the wave groups, initially in deep water, propagate up a constant slope until reaching shallow water. The overall comparison between the multi-layer model and the experiment is quite good, indicating that the multi-layer theory has good nonlinear, as well has linear, accuracy for deep-water waves.

scholarly journals X-type Soliton, Resonance Y-type Soliton and Hybrid Solutions of a (2+1)-dimensional Hirota-Satsuma-Ito System for the Shallow Water Waves

2021 ◽  
Author(s):  
Yuan Shen ◽  
Bo Tian ◽  
Tian-Yu Zhou ◽  
Xiao-Tian Gao
Keyword(s):  
Shallow Water ◽  
Symbolic Computation ◽  
Water Waves ◽  
Water Wave ◽  
Soliton Solutions ◽  
Hirota Method ◽  
Shallow Water Waves ◽  
Soliton Resonance ◽  
Hybrid Solutions

Abstract Water waves are observed in the rivers, lakes, oceans, etc. Under investigation in this paper is a (2+1)-dimensional Hirota-Satsuma-Ito system arising in the shallow water waves. Via the Hirota method and symbolic computation, we derive some X-type and resonance Y-type soliton solutions. We also work out some hybrid solutions consisting of the resonance Y-type solitons, solitons, breathers and lumps. Graphics we present reveal that the hybrid solutions consisting of the resonance Y-type solitons and solitons/breathers/lumps describe the interactions between the resonance Y-type solitons and solitons/breathers/lumps, respectively. The obtained results rely on the water-wave coefficient in that system.

Spectral data for travelling water waves: singularities and stability

2009 ◽  
Vol 624 ◽  
pp. 339-360 ◽  
Author(s):  
DAVID P. NICHOLLS
Keyword(s):  
Shallow Water ◽  
Spectral Data ◽  
Deep Water ◽  
Water Waves ◽  
Water Wave ◽  
Perturbation Expansion ◽  
Spectral Stability ◽  
Perturbation Approach ◽  
First Eigenvalue ◽  
Water Wave Problem

In this paper we take up the question of the spectral stability of travelling water waves from a new point of view, namely that the spectral data of the water-wave operator linearized about fully nonlinear Stokes waves is analytic as a function of a height parameter. This observation was recently made rigorous by the author using a boundary perturbation approach which is amenable to approximation by a stable high-order numerical method. Using this algorithm, we investigate, for both super- and sub-harmonic disturbances, the evolution of the spectrum, in particular the ‘first collision’ of eigenvalues and the ‘smallest singularity’ in the perturbation expansion. The former is studied in response to MacKay & Saffman's (1986) work on the water-wave problem which demonstrated that instability can only arise after the collision of two eigenvalues of opposite Krein signature. However, we present results which show, quite explicitly, that eigenvalue collision (even of opposite Krein signature) is insufficient to conclude instability. With this in mind, we have identified a new criterion for the loss of spectral stability, namely the appearance of a singularity in the expansion of the spectral data (as a function of the height parameter mentioned above). We give some heuristic reasons why this should be so, and then provide complete numerical spectral stability results for four representative depths, two above (h = ∞, 2) and two below (h = 1, 1/2) Benjamin's (1967) critical value, hc ≈ 1.363, above which the Benjamin–Feir instability emerges. We find that the strongest (two-dimensional) instability appears to be among the long waves, but we notice that there is a sharp difference between ‘shallow-water’ and ‘deep-water’ waves in that first eigenvalue collision and smallest expansion singularity are synonymous for shallow water, while this is not so in deep water where ‘windows of stability’ beyond the first eigenvalue collision exist.

scholarly journals Water Waves and Light: Two Unlikely Partners

2021 ◽  
Author(s):  
Georgios N. Koutsokostas ◽  
Theodoros P. Horikis ◽  
Dimitrios J. Frantzeskakis ◽  
Nalan Antar ◽  
İlkay Bakırtaş
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Soliton Solutions ◽  
Type Equation ◽  
Beam Propagation ◽  
Generic Model ◽  
Two Dimensional ◽  
Shallow Water Waves ◽  
Nonlocal Nonlinear

We study a generic model governing optical beam propagation in media featuring a nonlocal nonlinear response, namely a two-dimensional defocusing nonlocal nonlinear Schrödinger (NLS) model. Using a framework of multiscale expansions, the NLS model is reduced first to a bidirectional model, namely a Boussinesq or a Benney-Luke-type equation, and then to the unidirectional Kadomtsev-Petviashvili (KP) equation – both in Cartesian and cylindrical geometry. All the above models arise in the description of shallow water waves, and their solutions are used for the construction of relevant soliton solutions of the nonlocal NLS. Thus, the connection between water wave and nonlinear optics models suggests that patterns of water may indeed exist in light. We show that the NLS model supports intricate patterns that emerge from interactions between soliton stripes, as well as lump and ring solitons, similarly to the situation occurring in shallow water.

Shallow water wave generation by unsteady flow

1968 ◽  
Vol 31 (4) ◽  
pp. 779-788 ◽  
Author(s):  
J. E. Ffowcs Williams ◽  
D. L. Hawkings
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Small Amplitude ◽  
Point Of View ◽  
Sound Waves ◽  
Aerodynamic Sound ◽  
Shallow Layer ◽  
Sound Theory ◽  
Water Simulation

Small amplitude waves on a shallow layer of water are studied from the point of view used in aerodynamic sound theory. It is shown that many aspects of the generation and propagation of water waves are similar to those of sound waves in air. Certain differences are also discussed. It is concluded that shallow water simulation can be employed in the study of some aspects of aerodynamically generated sound.

New integrable (2+1)- and (3+1)-dimensional shallow water wave equations: multiple soliton solutions and lump solutions

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Lump Solutions ◽  
Content Type ◽  
Shallow Water Wave ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Shallow Water Wave Equations

Purpose This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space. Design/methodology/approach The author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense. Findings The paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium. Research limitations/implications The author addresses the integrability features of this model via using the Painlevé analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method. Practical implications The obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition. Social implications The work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions. Originality/value The paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.

scholarly journals “Extraordinary” modulation instability in optics and hydrodynamics

2021 ◽  
Vol 118 (14) ◽  
pp. e2019348118
Author(s):  
Guillaume Vanderhaegen ◽  
Corentin Naveau ◽  
Pascal Szriftgiser ◽  
Alexandre Kudlinski ◽  
Matteo Conforti ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Stability Analysis ◽  
Linear Stability ◽  
Nonlinear Theory ◽  
Linear Stability Analysis ◽  
Water Waves ◽  
Modulation Instability ◽  
Nonlinear Effects ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

The classical theory of modulation instability (MI) attributed to Bespalov–Talanov in optics and Benjamin–Feir for water waves is just a linear approximation of nonlinear effects and has limitations that have been corrected using the exact weakly nonlinear theory of wave propagation. We report results of experiments in both optics and hydrodynamics, which are in excellent agreement with nonlinear theory. These observations clearly demonstrate that MI has a wider band of unstable frequencies than predicted by the linear stability analysis. The range of areas where the nonlinear theory of MI can be applied is actually much larger than considered here.

Multiple soliton, fusion, breather, lump, mixed kink-lump and periodic solutions to the extended shallow water wave model in (2+1)-dimensions

2020 ◽  
pp. 2150138
Author(s):  
Hajar F. Ismael ◽  
Aly Seadawy ◽  
Hasan Bulut
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Soliton Solutions ◽  
Quadratic Function ◽  
Wave Model ◽  
Simple Method ◽  
Shallow Water Wave ◽  
Hirota Bilinear Form ◽  
The One ◽  
Physical Phenomena

In this paper, we consider the shallow water wave model in the (2+1)-dimensions. The Hirota simple method is applied to construct the new dynamics one-, two-, three-, [Formula: see text]-soliton solutions, complex multi-soliton, fusion, and breather solutions. By using the quadratic function, the one-lump, mixed kink-lump and periodic lump solutions to the model are obtained. The Hirota bilinear form variable of this model is derived at first via logarithmic variable transform. The physical phenomena to this model are explored. The obtained results verify the proposed model.

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